Johnson is by far the most effective president we've ever had in dealing with the Congress.

Johnson was as great an executive legislator as ever sat in the White House. He had largely learned his craft as Senate majority leader, where he had cultivated the art of exchanging favors for votes.

(Brinkley and Dyer, Eds., The American Presidency, p. 414).

LBJ was tremendously insecure and tremendously ambitious, and while I'm certain he had prejudices, he looked past them to benefit poor Americans and Black Americans. Partly out of self-interest, as he saw lifting them from poverty would be his greatest legacy, but he also saw lifting people from poverty as a justifiable end to whatever means he saw to pursue.

Johnson was obsessed with politics and it was his only hobby. John Connally, Johnson friend and former governor of Texas, said that they'd go to baseball games together and not talk about the game, just politics. It was an obsession. When LBJ left office, he would have meetings with his ranch workers in the morning to go over the tasks of the day, just so he could have some sort of similarity to his staff meetings (Both from LBJ, the documentary from American Experience).

LBJ knew how the Senate worked not only because of his time as party leader, but he was also a party whip, and he also served in the House. He was known for "the treatment," and no discussion of LBJ, academic or otherwise, is complete without its mention. He would charm, he would threaten, he would harass, he would wine and dine, he would bargain. He would do anything he needed to do to get a vote on his side.

Here's a great example. LBJ is on the phone with Rep. Albert Thomas trying to get language removed from a bill that would force the president to report wheat sales to the Soviet Union.

LBJ was not at all refined and he had no polish as a politician, but he knew how to talk to people in a very human way.

We've had former presidents serve in the Congress before, but none of them were able to control the ins-and-outs of the legislature like LBJ, and that understanding helped him a great deal while president. It's also a reason why Johnson was so miserable as vice president, as that office had absolutely zero influence on any sort of policy.

Edit: misspelling

I don't have the time or the background to do a statistical analysis on whether you can consider Lyndon Johnson an "outlier" in terms of the success of his congressional relations efforts. But I can make the case that he had a remarkable talent for persuasion (both in public and in private), could cultivate important allies, and implement effective strategies. He developed these skills over the course of his lengthy public service career, and it would serve him well when he reached the White House. In my view, his reputation as an effective legislator is well-deserved. In discussing the skills necessary to be effective with Congress  Joseph A. Pika et. al., wrote in their book The Politics of the Presidency

No president has ever possessed all of these skills. By most accounts, however, Lyndon Johnson exhibited more than any other modern president, and made the most effective use of his involvement in the legislative process.

LBJ had a gift for persuasion throughout his life. In 1928, he was a university student in San Marcos, Texas. In June of that year, the Democratic Party held its Convention in Houston to nominate Al Smith for President. Johnson's college had a rule prohibiting students from driving outside the San Marcos area, but he convinced a friend to drive him to Houston anyway. Once there, he brandished copies of the the College Star (the student paper) and talked his way into the convention claiming to be a member of the press. When he got back, Dean Alfred Nolle summoned him for a disciplinary meeting over the forbidden trip, but Johnson was so animated about the whole thing that he was able to talk his way out of trouble. Even before he showed up in Congress, he had a gift for getting people to see things his way.

Indeed, it was thanks in part to this ability that he made it to Washington in the first place. In 1930, the young Johnson gave a speech at a local fair on behalf of a politician named Pat Neff, who was running for election to the State Railroad Commission. Neff had once done a favour to Johnson's father, so he gave a good speech for him, which impressed the crowd, among which was a man named Wally Hopkins, who was running for the State Senate. Hopkins put Johnson on his campaign, and later recommended him to newly elected Texas Congressman Richard Kleberg.

Johnson went to work in Kleberg's office in 1931 and immediately began immersing himself in Washington politics. Congressional staffers at the time had their own debating society, commonly known as "The Little Congress," and Johnson soon became one of its most influential members. He befriended future House Majority Leader (and future Speaker) Sam Rayburn, and the two developed a close relationship. When Johnson wanted to become Director of the Texas branch of the newly created National Youth Administration (NYA) in 1935, Rayburn put in a good word for him with Texas Senator Tom Connally, who passed this on to President Roosevelt, who ultimately gave him the job.

Johnson secured his own election to the House in 1937, winning a special election in the 10th District after the death of incumbent rep. James P. Buchanan. Over the course of his career in the House, he developed a good working relationship with President Roosevelt, something he was able to put to good use in bringing rural electrification to his constituents. The Rural Electrification Administration approved a loan to an electrical co-operative in Johnson's district after he made a presentation to the President himself.

Outside of his time in the White House and as Vice President, LBJ is of course very well known for his time in the Senate, particularly his tenure as majority leader from 1955 - 1961. Much like everything else, I've touched on in this answer, that's worth a lengthy analysis on its own, but I'll elaborate a bit here. Johnson became leader of the Senate Democrats after a quick ascension up the ranks. He won election to the Senate in 1948, became Democratic Whip in 1951, then leader in 1953 after previous leader Ernest Macfarland lost his seat in 1952. He secured the leadership thanks to the support of Richard Russell of Georgia and Earle Clements of Kentucky, allies he had effectively cultivated.

It's far too in-depth for a forum like this, but Johnson's 1954 strategy against Senator John Bricker and his isolationist proposal to amend the Constitution is a great example of his ability to manage Senate proceedings. See here for a great overview. Another example is Johnson's management of the 1957 Civil Rights Act. Securing its passage earned Johnson credit but he of course put together a compromise that watered down the already modest law, and even then only secured enough votes by promising Senators from the West Southern support for the Hells Canyon Dam project in Idaho. As a simple piece of legislative maneuvering though, it was a masterstroke.

By the time he became President, LBJ had come to know Washington exceptionally well. His well-known work on the Civil Rights Act of 1964 is of course only part of a broad effort across Congress and civil rights activists, but his contribution can't be discounted. It's worth noting that Richard Russell, his onetime patron and later ardent foe on civil rights, once said "You know, we could have beaten John Kennedy on civil rights, but not Lyndon Johnson."

Johnson's lengthy career in Washington before he became President meant that he knew how to manage congressional relations and to achieve significant legislative successes. His administration's work with Wilbur Mills of Arkansas, Chair of the House Ways and Means Committee helped the Social Security Amendments of 1965 succeed in the House, while his effective pressure on Harry Byrd contributed to victory in the Senate, not to mention a good working relationship with Majority Leader Mike Mansfield, which would also be important in securing passage of the Voting Rights Act.

In this answer, I've focused on Johnson's pre-White House career, which I know wasn't what this question wanted addressed. However, I think that by looking at his time in Congress, we can see how he became so effective, which in turn sheds some light on how he was able to put his Congressional skills to good use once he became President.